Two point sources emit sound waves of 1 m wavelength. The sources are pointed at each other. The sources, 2 m apart, emit waves which are in phase with each other at the instant of emission.
a) Where, along the line between the sources are the waves out of phase with each other by pi radians?
b) If the emitted wavelength changes to 0.5 m, where are the waves out of phase with each other by pi/2 radians.
To solve these problems, we need to understand the concept of path difference between two waves. Path difference is the difference in distance traveled by two waves from their sources to a specific point.
a) To find the location where the waves are out of phase by π radians (180 degrees), we need to determine the path difference, given a wavelength of 1 m.
Step 1: Calculate the path difference:
Since the two sources are pointed towards each other, we can consider the midpoint between them as the reference point.
The path difference, Δx, can be calculated using the formula: Δx = nλ/2
Here, n is an integer, and λ is the wavelength.
Since we are interested in the point where the waves are out of phase by π radians, we can assume n = 1.
Δx = (1 * 1 m) / 2 = 0.5 m
Step 2: Determine the location of the out-of-phase point:
The out-of-phase point will be located at a distance of 0.5 m from the midpoint between the sources, towards one of the sources.
Therefore, the waves are out of phase by π radians at a point located 0.5 m away from the midpoint on the same side as one of the sources.
b) To find the location where the waves are out of phase by π/2 radians (90 degrees), we need to determine the new path difference when the wavelength changes to 0.5 m.
Step 1: Calculate the new path difference:
Using the same formula, Δx = nλ/2, we substitute the new wavelength, λ = 0.5 m.
Δx = (n * 0.5 m) / 2
Since we are looking for π/2 radians phase difference, n = 2.
Δx = (2 * 0.5 m) / 2 = 0.5 m
Step 2: Determine the location of the out-of-phase point:
Similar to the previous question, the out-of-phase point will be located at a distance of 0.5 m from the midpoint between the sources, towards one of the sources.
Therefore, the waves are out of phase by π/2 radians at a point located 0.5 m away from the midpoint on the same side as one of the sources.